Past exam of the condensed matter physics major option course of the University of Oxford / 2015 / 8

Start from a gauge-invariant Ginzburg-Landau theory of superconductivity gradient energy $|(-i\hslash \nabla -q\mathbf{A})\psi {|}^2/(2m)$, with $\psi$ the coherent pair order parameter and $q$ its charge. Varying the free energy with respect to $\mathbf{A}$ gives the current density displayed in the source. Write $\psi =\sqrt{n_s}{e}^{i\theta }$ to obtainSingle-valuedness around a loop in superconducting material requires $\oint \nabla \theta \cdot d\mathbf{l}=2\pi n$. Integrating the current relation therefore gives the exact fluxoid conditionFor a thick ring choose the contour sufficiently deep in the material that the current term is negligible. Then flux itself is approximately quantized, with positive quantumThe sign of the pair charge is absorbed into the orientation of $n$. In a thin ring with appreciable kinetic inductance, it is the fluxoid, not exactly the geometrical flux alone, that remains quantized. A state with nonzero phase-gradient current is a concrete counterexample to dropping the current integral without justification. The flux-only identity thus uses the ordinary thick-ring or negligible-kinetic-inductance limit in addition to phase single-valuedness.

Cool a thick superconducting ring through its transition in a controlled external flux, remove or change the applied field while keeping the ring superconducting, and measure the persistent trapped flux using a calibrated pickup coil and sensitive magnetometer. Repeating while varying the cooling flux selects successive winding sectors. The measured flux is close to steps separated by ${\Phi }_0$ when the kinetic-inductance correction is negligible. A SQUID-based flux readout can resolve that spacing; scanning local magnetic imaging offers another way to observe the trapped flux without identifying the ring current with an ordinary dissipative transport current.

The experiment tests the phase-winding/fluxoid relation in a. Self-inductance, London penetration depth and the size of the ring determine how nearly the measured geometric flux follows integer steps. A periodic response alone should not be used to infer an absolute trapped flux without calibration and the relevant inductive corrections.

TODO — source figure gap: the surviving ordinary-ring and two-junction-ring outlines and labels are readable, but page 9's figure form calls 52 absent raster objects, /R514 through /R565. The original archived parent has identical damage. An intact figure is needed to recover any missing image-specific details; those raster components have not been inferred or recreated. The observation method above answers the physical request without requiring the unavailable artwork; the TODO concerns verification of the input illustration rather than a missing part of the solution. The original circuit schematic used in c is an independent diagram of the stated model, not a claimed reconstruction of the unavailable artwork.

Let ${\delta }_1$ and ${\delta }_2$ be gauge-invariant phase drops across the two identical junctions, oriented between the same terminal phases. Their currents obey the DC Josephson effect: $I_1=I_C\sin {\delta }_1$ and $I_2=I_C\sin {\delta }_2$. For negligible loop self-inductance and no additional phase drops in the arms, fluxoid consistency fixes their difference modulo $2\pi$:Define the mean terminal phase $\delta =({\delta }_1+{\delta }_2)/2$ and $\varphi =\pi \Phi /{\Phi }_0$. Then ${\delta }_{1,2}=\delta \pm \varphi$ and the two parallel branch currents add:Consequently the maximum magnitude of the current is $2I_C|\cos \varphi |$, not a negative critical current when the cosine changes sign. The phase shifts by $\pi$ between such stable branches. Junction asymmetry or appreciable self-generated flux modifies the simplest interference pattern, so the stated idealization is required before treating $\Phi$ as an externally prescribed flux.

The AC Josephson effect is $V=({\Phi }_0/2\pi )\dot{\delta }$. Since the imposed flux is held fixed when defining a small-signal inductance, differentiate c to obtainOn the stable small-current branch with $\cos \varphi >0$, $\delta \simeq 0$ and the result reduces to the printed expression,The stable small-signal inductance of a symmetric DC SQUID requires $|I|\ll 2I_C|\cos \varphi |$, stronger than $I\ll I_C$ near a half-integer flux quantum. If $\cos \varphi <0$, the stable zero-current phase is $\delta \simeq \pi$ and $\cos \delta \simeq -1$. Its inductance is positive:For example, $\Phi ={\Phi }_0$ makes the printed signed formula negative when expanded about $\delta =0$, but that phase is an energy maximum. Expanding about the stable $\delta =\pi$ corrects the sign. At ideal half-integer flux the critical current vanishes and the linear inductance formally diverges; a finite transport current cannot satisfy the small-current approximation there. Geometric inductance and other nonidealities then have to be retained instead of extending the simple formula through its pole.